Using Ray-shooting Queries for Sublinear Algorithms for Dominating Sets in RDV Graphs

Therese Biedl, Prashant Gokhale

Abstract:

In this paper, we study the dominating set problem in RDV
graphs, a graph class that lies between interval graphs and chordal graphs
and defined as the vertex-intersection graphs of downward paths in a
rooted tree. It was shown in a previous paper that adjacency queries
in an RDV graph can be reduced to the question whether a horizontal
segment intersects a vertical segment. This was then used to find a max
imum matching in an n-vertex RDV graph, using priority search trees,
in O(nlogn) time, i.e., without even looking at all edges. In this paper,
we show that if additionally we also use a ray shooting data structure,
we can also find a minimum dominating set in an RDV graph O(nlogn)
time (presuming a linear-sized representation of the graph is given). The
same idea can also be used for a new proof to find a minimum dominating
set in an interval graph in O(n) time.